The problem with out-of-order viewing isn't so much spoilers as coherence. By and large, things will make the most sense if viewed in production order (not necessarily the same as broadcast/release order). For example, TOS "The Cage" could be the ideal place to start because it was the very first episode written/produced, even though it was never aired as such (but was cut into the later two-part "The Menagerie"), and only presented in its original form decades later. Likewise, much of what eventually became the Star Trek canon as we know it evolved (in production order) over the first 8 or 10 episodes of the first TOS season. Watching them in broadcast order (as everyone would have back in 1966) might seem a bit confused; sets, costumes (uniforms), Spock's look and demeanor, terminology, etc. all evolved and took until about the 10th episode to crystallize.
TNG had some minor character development over its run (one major cast change approaching the end of the first season), and some stories reference events from earlier episodes, so would make the most sense to watch in order. Also, the very first and very last episodes (both 2-parters) bookend the series, so should be watched in series order.
DS9 had major arcs in its later seasons, and also had bookending first and last episodes.
Voyager can be watched out of order, excepting for the first episode which establishes the show and the last episode which, again serves as a bookend. There are no story arcs, however there are several character arcs which will make more sense when the show is watched in broadcast order, especially a major casting change which occurs in the two-part cliffhanger bridging seasons 3&4.
Enterprise's third season has a major arc, and should be watched in order. Again, the very first and last episodes of the series are bookends. Watching the entire series in order can offer one reward - seeing the familiar Trek technology and other conventions build up a bit at a time.
As for the movies: #3 (Search for Spock) builds its story on the ending of #2 (Wrath of Khan). Although #4 (Voyage Home) tells a stand-alone story, its beginning starts with the state at which #3 ends. This makes movies 2,3, and 4 a sort of loose trilogy.